Nonlinear thermal radiation and the slip effect on a 3D bioconvection flow of the Casson nanofluid in a rotating frame via a homotopy analysis mechanism

Nomenclature

u , v , w

velocity components

Ω

dimensional rotational parameter

ρ m

density of the microorganism

T w , C w

surface temperature and concentration

T ∞ , C ∞

surrounding temperature and concentration

C p

specific heat at constant pressure

σ

electrical conductivity

ρ

density of the fluid

k f

thermal conductivity of the nanofluid

μ f

dynamic viscosity of the nanofluid

β

Casson parameter

D B

Brownian diffusion coefficient

D T

thermophoretic coefficient

k *

Porosity parameter

K 1

dimensional slip parameter

h f

convective heat transfer coefficient

Pe

Peclet number

A

slip parameter

δ 1

concentration difference parameter of microorganisms

Bi

Biot number

ω

rotation parameter

N b

Brownian motion parameter

N t

thermophoretic parameter

Sc

Schmidt number

R d

radiation parameter

λ

mixed convection parameter

M

magnetic parameter

N c

Rayleigh number

N r

buoyancy ratio parameter

θ w

temperature ratio parameter

Pr

Prandtl number

Le

Lewis number

1 Introduction

Nanofluids (NFs) are a type of heat transfer fluids that consist of a base fluid and small particles, typically ranging from 1 to 100 nm in size, which are suspended within the fluid. These particles are usually prepared by materials such as metals, metal oxides, or carbon nanotubes. NFs have been shown to have several applications in various industries, including heat transfer (see a review by Panduro et al. [1]), where NFs have significantly higher thermal conductivity than their base fluids [2], making them useful for applications where high heat transfer rates are required [3], such as in cooling systems for electronics, engines, and power plants [4,5]. Solar energy is another application where NFs can be utilized to develop the efficiency of solar thermal collectors by increasing the heat transfer from the collector to the fluid, thereby improving the overall energy conversion efficiency (see a review work of Shah and Ali [6]). Biomedical engineering is the recent application of NFs, where NFs have potential applications in drug delivery [7] and hyperthermia cancer therapy [8], where they can be used to target specific cells or tissues. Environmental remediation includes the application of NFs, whenever NFs can be used to remove contaminants from soil and groundwater by enhancing the rate of chemical reactions or physical processes such as adsorption (see Sonawane et al. [9] and Alsaady et al. [10] for an advanced study). Lubrication NFs can be used as lubricants in various industrial applications, including aerospace and automotive industries, due to their ability to reduce friction and wear [11]. Electronic cooling in NFs can be used as coolants in electronics to improve their heat dissipation efficiency, leading to longer lifetimes and increased performance [12]. Overall, NFs have the potential to revolutionize various industries by offering improved heat transfer and other properties compared to traditional fluids. Khan et al. [13] proposed the mechanical aspects of Maxwell NF flow in a dynamic system with entropy generation. However, further research is needed to fully understand their behavior and optimize their performance in specific applications.

The Casson NF model (CNFM) is a rheological model that describes the behavior of fluids with a yield stress, such as NFs. NFs are suspensions of nanoparticles in a base fluid, typically water or oil, and have unique thermal and rheological properties compared to their base fluids [14]. Vasefi et al. [15] studied the turbulent forced convective heat transport of CuO/water NF flow using an artificial intelligence network in a square duct. Gul et al. [16] scrutinized the simulation of water-based hybrid NF flow over a porous cavity with heat transfer. The solution to the flow problem is acquired numerically through the control volume method.

The CNFM assumes that the fluid behaves as a Bingham plastic, which means that it has a yield stress below which it does not flow, and above which it behaves like a Newtonian fluid with a constant viscosity [17, 18]. For NFs, the CNFM can be modified to include the effects of the concentration and size of nanoparticles. The viscosity and yield stress of NFs typically increase with increasing nanoparticle concentration, while the size of the nanoparticles can affect the yield stress but has little effect on the viscosity [19]. The CNFM is commonly used to predict the rheological properties of NFs and to optimize their use in various applications, such as heat transfer and liquid flowing in a microchannel [20,21]. However, it should be noted that the model has limitations, and more complex models may be required for accurately predicting the behavior of NFs under certain conditions [22,23].

Gyrotactic microorganisms (GMs) are microorganisms that have the ability to orient themselves in the direction of gravity. They are able to sense gravity and use it to move up or down in a fluid environment. Gyrotaxis is the process by which these microorganisms orient themselves with respect to the gravitational field. GMs can be found in a variety of aquatic environments, including oceans, lakes, and rivers. Examples of GMs include certain types of plankton, algae, and bacteria (see Hosseinzadeh et al. [24]). The ability of GMs to move in response to gravity has important ecological implications [25]. For example, in aquatic environments with vertical gradients in nutrient and light availability [26], gyrotaxis allows these microorganisms to position themselves under optimal conditions for growth and survival [27]. In addition, gyrotaxis can also affect the vertical distribution of microorganisms in aquatic ecosystems, with implications for nutrient cycling and carbon sequestration [28]. The study of GMs has applications in fields such as ecology, environmental science, and microbiology. It can also have practical applications in areas such as bioremediation, where GMs can be used to remove contaminants from water or soil [29]. Zuhra et al. [30] proposed the simulations of bioconvection second-grade NF flow with the suspension of nanoparticles containing GMs. For simulation, the optimal homotopy analysis method is utilized to evaluate the flow problem. Majeed et al. [31] explored the 2D bioconvection magnetized Casson nanoliquid flowing with Brownian and thermophoretic phenomena containing GMs. The thermal examination of a radiative bioconvective magnetized flow including GMs with the energy of activation was shown by Majeed et al. [32]. Zeeshan et al. [33] scrutinized the semi-analytical method with the DTM-Pade approach in a squeezing flow between two circular plates.

3D rotating geometry (3D-RG) refers to the study and manipulation of three-dimensional shapes and objects as they rotate around an axis or pivot point. This concept is commonly used in fields such as computer graphics, animation, and engineering. In computer graphics and animation, 3D-RG is often used to create realistic-looking objects and scenes. By applying mathematical formulas and algorithms to 3D models, it is possible to simulate the way an object would move and rotate in real life. This is used to create dynamic visual effects such as spinning logos, rotating planets, and flying objects. In engineering, 3D-RG is used to design and analyze complex structures such as gears, turbines, and engines. By modeling these structures in 3D and simulating their rotation, engineers can test their performance under different conditions and identify potential problems before building a physical prototype. Overall, 3D-RG is a powerful tool for visualizing and manipulating three-dimensional objects and structures, and it has a wide range of practical implementations in various disciplines. When a 3D object rotates in a fluid, it creates a disturbance in the fluid flow around it [34]. The fluid will experience forces due to the rotation of the object, which can cause the fluid to move in different ways. This phenomenon is known as fluid–structure interaction. The behavior of the fluid around the rotating object depends on diverse considerations, such as the speed of rotation, the size and shape of the object, the viscosity of the fluid, and the properties of the fluid itself [35]. One example of 3D-RG in a fluid is a propeller in water. As the propeller blades rotate, they create a flow of water around them. This flow of water produces forces that propel the boat forward (see Juhany et al. [36]). The design of the propeller blades is crucial for optimizing the efficiency of this process. Another example is a wind turbine. As the blades of the turbine rotate, they interact with the air around them, creating a flow of air that generates electricity [37]. In summary, when 3D-RG interacts with a fluid, it creates disturbances in the fluid flow around it, which can cause the fluid to move in different ways. The behavior of the fluid depends on various factors, and understanding these factors is crucial for optimizing the performance of rotating machinery in fluid environments.

Thermal radiative fluxing is the emission of electromagnetic waves by matter that is at a temperature higher than absolute zero. The rate at which thermal radiation is emitted depends on the temperature of the emitting object and the emissivity of the material. In NFs, the nanoparticles can increase the emissivity of the fluid, which leads to a higher rate of thermal radiation emission (TRE) [38,39]. The increased TRE in NFs can have several effects on the thermal properties of the fluid. For example, it can improve the heat transference rate between the fluid and a heat source or sink. It can also influence the temperature distribution within the fluid, which can impact the fluid flow behavior [40]. However, the exact mechanisms by which TRE affects the thermal properties of NFs are still being studied. Some studies have suggested that the size, shape, and concentricity of the nanoparticles can affect the emissivity of the fluid and therefore the rate of TRE [41]. Other studies have examined the impact of TRE on the thermal conductivity of the fluid, which can also be influenced by the nanoparticle properties [42,43]. In summary, thermal radiation is an important factor in the thermal properties of NFs [44]. Its effects can lead to enhanced heat transfer and fluid behavior; however, much research is needed to fully understand the mechanisms involved.

The homotopy analysis methodology (HAM) is a great analytical procedure used in the study of nonlinear problems in various fields, including fluid mechanics. It was first introduced by J.H. He in 1999 and has since been widely used in many scientific and engineering applications. In fluid mechanics, the HAM can be directed to solve a broad collection of challenges, including laminar and turbulent flow (see Ige et al. [45]), boundary layer flow, and free surface flow. The method involves constructing a homotopy between a known linear solution and the nonlinear solution of interest [46]. This allows the nonlinearity of the problem to be gradually introduced and solved systematically [47]. The HAM can also be used to study the stability of fluid flows, as well as the bifurcation behavior of fluid systems. It has been successfully applied to investigate complex phenomena such as chaos and turbulence in fluid mechanics [48]. One of the main advantages of the HAM is its ability to provide closed-form solutions for nonlinear problems that are difficult or impossible to solve using traditional analytical methods [49]. Khan et al. [50] investigated the thin film flow of a second-grade fluid past a stretching sheet with heat transfer in a porous medium. The solution of the model is acquired using the HAM. Ahmad et al. [51] studied the MHD thin film flow of the Oldroyd-B fluid with bioconvection phenomenon and activation energy using the HAM. This makes it a useful tool for researchers and engineers working in fluid mechanics and related fields. Generally, the HAM is a promising approach for the study of nonlinear problems in fluid mechanics, and its potential for solving complex problems and providing valuable insights into fluid behavior makes it an important tool for researchers and engineers in this field.

The previous theoretical investigations point to a previously non-explored nonlinear thermal radiation and the applied magnetic feature-based three-dimensional Casson nanomaterial flow in porous media. In the design of the rotating frame, the flow has been assumed. Bioconvection flows are intimately connected to mechanical and real-life phenomena, such as the construction and design of biological cells, biological conjugates, and microscopic biological systems, and they have emerged as a hot issue in present-day scientific research because of this close relationship. To research the applications of bioconvection, GMs are used in the Casson NF. In order to include the thermophoretic and Brownian phenomena, the Buongiorno thermal nano-model has been modified. In order to resolve the flow issue, convective boundary conditions have to be circumvented. It is possible to generate the dimensionless pattern of equations if appropriate variables are used. Subsequently, the answers to the nonlinear formulations are derived by the use of an analytical technique called the homotopy analysis mechanism (HAM).

2 Mathematical formulation

This article centers on a 3-D Casson NF investigation containing rotating GMs. A cartesian system places the surface in the xy -plane and the fluid in area z ≥ 0 . An elastic sheet is stretched with an angular velocity Ω along the x-axis, as shown in Figure 1. The initiation energy relatives and nonlinear thermal pattern are taken into consideration in the energy and concentration equations, respectively. In a velocity field, u and v stand for the normal and horizontal velocity components, respectively. Using these flow possessions, the main equations for the Casson NF with motile microbes are demarcated as tracks [52,53]. In this article, the following presumptions are made:

1. The velocity components along the x, y, and z directions are u , v and w separately.

2. The coordinate system is assumed in such a way that z-axis is supposed to be orthogonal to the sheet, whereas the x- and y -axis are chosen sideways with the plate. Continuous distortion of the plate is measured along the x- and y -axis with velocities u w ( x ) and v w ( y ) , respectively.

3. Because the flow is generated by a bilinear stretched surface, the pressure gradient along the sheet is ignored.

4. While viscous dissipation is taken into account, velocity slippery with a dimensional slippery factor K 1 and non-linear thermal radiation are not taken into consideration.

5. The Buongiorno thermal nano-model is used with the suspension of GMs.

6. The NF flow is incompressible and steady in a rotating system.

Figure 1

Flow geometry of the problem.

The justified boundary conditions for the current analysis are

where β is the Casson parameter, ν is the kinematics viscosity, ρ f is the density of the NFs, k * is the porosity parameter, g ∼ is the gravitational acceleration, Ω denotes the dimensional rotational parameter, β g is the volume expansion coefficient, ρ p is the density of the nanoparticles, ρ m is the density of the microorganism, T is the temperature, T w and T ∞ , respectively, represent the surface and surrounding temperatures, C w and C ∞ , respectively, represent the surface and surrounding concentrations, α m = k ( ρ c ) f is the thermal diffusivity (with k being the thermal conductivity), ( ρ c ) f is the heat capacity of the fluid, ( ρ c ) p is the heat capacity of the nanoparticles, h f represents the convective heat transfer coefficient, K 1 denotes the dimensional slip parameter, D B stands for the Brownian diffusion coefficient, and D T is the thermophoretic coefficient.

3 Method of solution

To simplify the flow governing equations and the associated boundary conditions, we introduce the following similarity transformations:

It is evident that the similarity variables described in equation (8) satisfy equation (1) in the same way. Now, one can obtain this by applying equation (8) to equations (2)–(7):

The transformed boundary conditions are

The parameters are the rotation parameter ω = Ω a , the mixed convection parameter λ = β g g ∼ T ∞ ( 1 − C ∞ ) a u w , the magnetic parameter M = σ B 0 2 a ρ f , the Rayleigh number N c = γ g ∼ T ∞ ( ρ m − ρ f ) ( n w − n ∞ ) β g ρ f ( 1 − C ∞ ) T ∞ , the buoyancy ratio parameter N r = ( ρ p − ρ f ) ( C w − C ∞ ) β g ρ f ( 1 − C ∞ ) T ∞ , the radiation parameter R d , the temperature ratio parameter θ w = T w T ∞ , θ w > 1 , the Prandtl number Pr , the Brownian constant N b = D B τ ( C w − C ∞ ) ν , the thermophoresis parameter N t = D T τ ( T w − C ∞ ) ν T ∞ , the Schmidt number Sc = ν D B , the Peclet number Pe = b w m D m , the concentration difference constant of the microorganism δ 1 = n ∞ n w − n ∞ , and the Lewis number Le = ν D m . Bi denotes the Biot number, and the slip parameter A = N 0 a ν . The skin friction coefficient C f and the local Nusselt number N u x , which are both physical variables of relevance, are defined as

The surface shear stress τ w , the surface heat flow q w , the wall mass flux q s , and wall motile microorganisms’ flux N n are determined as

By employing the similarity transformation (8), we obtain

where Re x = a x 2 ν f is the local Reynolds number.

4 HAM

Liao first proposed the homotopy strategy in 1992. The homotopic technique is preferred to find nonlinear model solutions. In contrast to prior perturbations and non-perturbative processes, the homotopy assessment approach enables us to quickly adjust and alter the convergence position and charges of learned guesses in response to changing circumstances. This is the case because it gives us the ability to examine homotopy. In a nutshell, the homotopy evaluation has the following additional advantages: it can be used to precisely make assumptions about a nonlinear mathematical challenge by choosing particular units of base functions; it is valid even if a given nonlinear problem does not contains either small or extensive parameters at all; and it can provide us with a convenient way to change and manipulate the convergence place and charge of estimation collection when it is necessary to do so.

4.1 Limitations of HAM

In the field of mathematics, HAM is a sophisticated approach that is used to solve nonlinear differential equations. On the other hand, just like any other approach, it does have certain drawbacks. HAM has several significant drawbacks, and some of which are listed as follows:

1) Convergence: The convergence of the HAM series solution is not guaranteed for all types of problems. In some cases, the series may not converge or may converge very slowly, leading to inaccurate or impractical results. The convergence behavior depends on the nature of the problem and the chosen auxiliary linear operator.

2) Solution dependence: The HAM solution is highly dependent on the initial approximation and the auxiliary linear operator. Different choices of these parameters can lead to different results. It can be challenging to determine the optimal values, and there is no systematic procedure for selecting them.

3) Computational complexity: The computational complexity of the HAM can be high, especially for problems with multiple nonlinear terms. The process of calculating the coefficients of the series solution can be time-consuming and require substantial computational resources.

4) Singularity problems: HAM may encounter difficulties when dealing with singularities, such as poles or essential singularities, in the domain of the problem. The presence of singularities can affect the convergence behavior of the series solution and require special treatment.

5) Limited applicability: HAM is primarily suitable for solving differential equations with small parameters or perturbation terms. It may not be as effective for problems with large nonlinearities or strong nonlinearity.

6) Lack of rigorous error analysis: HAM lacks a rigorous error estimation or error control mechanism. It can be challenging to assess the accuracy or reliability of the obtained solutions.

7) Limited extension to higher dimensions: While HAM has been successfully applied to one-dimensional problems, its extension to higher-dimensional problems is more challenging. The complexity and computational requirements increase significantly in higher dimensions.

Despite these limitations, the HAM remains a valuable tool for solving a wide range of nonlinear problems. Researchers continue to develop new variants and improvements to address some of these limitations and enhance the applicability of the method. Figure 2 shows the phases of the homotopic approach [54].

Figure 2

Methods for obtaining numerical results.

By using the HAM in Mathematica 13, an analytical solution to the higher-order ordinary differential equations may be obtained. The following are some definitions for the initial estimates and linear operators:

(18) f 0 ( η ) = 1 1 + A 1 + 1 β ( 1 − e − η ) , g 0 ( η ) = e − η , θ 0 ( η ) = Bi 1 + Bi e − η , ϕ 0 ( η ) = − NtBi Nb ( 1 + Bi ) e − η , χ 0 ( η ) = e − η .

(19) L f = f ‴ − f ′ , L g = g ″ − g , L θ = θ ″ − θ , L ϕ = ϕ ″ − ϕ , L χ = χ ″ − χ ,

with

(20) L f [ C 1 + C 2 exp ( η ) + C 3 exp ( − η ) ] = 0 , L g [ C 4 exp ( η ) + C 5 exp ( − η ) ] = 0 , L θ [ C 6 exp ( η ) + C 7 exp ( − η ) ] = 0 , L ϕ [ C 8 exp ( η ) + C 9 exp ( − η ) ] = 0 , L χ [ C 10 exp ( η ) + C 11 exp ( − η ) ] = 0 ,

where C i ( i = 1 − 11 ) are arbitrary constants.